What is an Identity Matrix?
The Identity Matrix is a multiplicative unit of the Matrix. Before learning what the Identity Matrix is, let's recall the importance of units in Mathematics. Identity is a Mathematical quantity, and when manipulated in a particular quantity, the same quantity remains. Consider the following Example.
The additive Identity element is 0. This is because adding any number to 0 gives the same number as the total. For Example, 3 + 0 = 3, 0 + (1) = 1. Multiplying any number by 1 yield the same number as the product, so the multiplication Identity is 1. For Example, 3 × 1 = 3, 1 × (1) = 1. Similarly, of course, if you add the zero Matrix to any 2x2 Matrix, you'll see that you get the same Matrix, or zero. The Matrix becomes an additive Identity Matrix called. But what is the multiplicative identity in terms of Matrices? Let's learn this in detail here.
The Identity Matrix is a Square Matrix in which each element of the main diagonal is 1 and each other element is 0. Also known as the Identity Matrix. Represents an Identity Matrix of degree n × n (or n) as I.
This is sometimes referred to simply as I.
Identity Matrix Properties
Let us discuss the properties of the Identity Matrix.
Identity Matrix is always in the form of a Square Matrix.
The Identity Matrix is called a Square Matrix because it has the same number of rows and columns. For any given whole number n, the Identity Matrix is given by n x n.
Multiplying a given Matrix with the Identity Matrix would result in the Matrix itself.
Since the multiplication is not always defined, the size of the Matrix matters when you work on the Matrix multiplication.
For Example, for the given m x n Matrix C, you get
\[ C = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8\end{bmatrix}\]
The above Matrix is a 2 x 4 Matrix since it contains 2 rows and 4 columns.
When multiplying two inverse Matrices, you would get an Identity Matrix.
If you multiply two Matrices that are inverses of each other you would get an Identity Matrix.
The properties of the Identity Matrix based on the definition are:
All Identity Matrices are Square Matrices. All identities are diagonal because only the elements of the main diagonal are non-zero.
All identities are scalar Matrices because all elements of the main diagonal are the same and all other elements are zero.
Multiply the Identity Matrix by another Matrix to get the same Matrix. According to the study, the Determinant of the Identity Matrix is 1.
The Identity Matrix is symmetric with respect to IT = I. The inverse of the Identity Matrix is itself I * I1 = I1 * I = I. In = I, integer 'n'.
That is if we count deeply the Square of the Identity Matrix is equal to itself, the cube of the Identity Matrix is equal to itself, and so on.
Multiply the Matrix by its reciprocal to get the Identity Matrix (see this in the next section). Find the inverse Matrix using the Identity Matrix
The inverse of Matrix A (denoted as A1) is Matrix B (and vice versa) only if AB = BA = I. Where A, B, and I are Square Matrices of the same degree. Given A and B, you can easily check that they are the opposite of each other by simply checking AB = BA = I. But given Matrix A, how can we find the inverse B? You can find the inverse of the Matrix by following the steps below.
Step 1: Use an Identity Matrix of the same degree to describe an augmented Matrix adjacent to the specified Matrix and separate these two Matrices with a line.
Step 2: Apply a row operation aimed at converting the left Matrix (A) to an Identity Matrix.
Step 3: The Matrix on the right and left is the inverse Matrix itself. The procedure for finding the inverse Matrix using the Identity Matrix is shown.
See the Identity Matrix Example section below for an Example of finding the inverse Matrix using these steps.
Identity Matrix Application
Identity Matrices are used in linear algebra for several purposes. The usage of the Identity Matrix is as follows: The
Identity Matrix is used to check if two given Matrices are opposite to each other. The Identity Matrix is also used to figure out the inverse of the Matrix. The Identity Matrix is used to find eigenvalues and eigenvectors. When solving simultaneous equations using basic row operations, the Identity Matrix is used. Important information about the Identity Matrix
There are some important points to keep in mind regarding the Identity Matrix.
If you see an Identity Matrix with no operations specified, it should be the Identity Matrix by default in terms of multiplication. To write an Identity Matrix of a particular order, first, write an empty Matrix of the specified order, write 1 in place of the main diagonal element, and finally write 0 in place of all other elements. If AB = BA = I, then A and B are opposite to each other. To find the inverse of a Matrix, write it next to the Identity Matrix of the same order on the right. Apply row operations to the entire expanded Matrix to create the Matrix on the left as an Identity Matrix. Then the Matrix on the right is the inverse of the specified Matrix.
The Identity Matrix is known as the Matrix that is in the form of the n × n Square Matrix in which the diagonal contains the ones and all the other elements are zeros. It is also referred to as a unit Matrix or an elementary Matrix. It is denoted as In or just I, wherein n is the size of the Square Matrix. To explain the Identity Matrix definition part by part, let us start by reminding you that the Square Matrix refers to the Matrix that contains the same amount of rows and columns. The order of the Matrix comes from its dimensions, and the main diagonal is the array of the elements that are inside the Matrix that form the inclined line starting from the top left corner and extending to the bottom right corner. Given the characteristics of the Identity Matrix, you can also conclude that these types of Matrices are also called diagonal Matrices. In this article, we will learn about what is an Identity Matrix, the Determinant of Identity Matrix, Identity Matrix properties, the Identity Matrix in c, and learn about the Identity Matrix Example.
Identity Matrix Definition
An Identity Matrix refers to a type of the Square Matrix in which its diagonal entries are equal to 1 and the off-diagonal entries are equal to 0.
Identity Matrices play a vital role in linear algebra. In particular, their role in the Matrix multiplication is similar to the role that is played by the number 1 when it comes to the multiplication of the real numbers:
The real number remains unchanged if it is multiplied by 1
The Matrix remains unchanged if it is multiplied by an Identity Matrix
Identity Matrix Example
Some Examples of Identity Matrices are as follows:
The 2 x 2 Identity Matrix is given by
\[ I = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\]
The Identity Matrix of order 3 is represented in the following manner:
\[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\]
Solved Examples
Example 1
Write an Identity Matrix of the order 4
Solution:
The Identity Matrix of the order 4 x 4 is given as
\[ I = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\]
Example 2
Determine if the given Matrix is an Identity Matrix or not.
\[ C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\]
Solution:
No, the given Matrix is not an Identity Matrix since it is not a Square Matrix. The number of rows is not equal to the number of columns. The given Matrix is of the order 2 x 3.
FAQs on Identity Matrix
1. What do you mean by an identity matrix?
Matrix multiplication is a type of a binary operation. Take two elements from a given set and then follow some rules and combine them together to some other element of the given set. Other examples of the binary operations include the addition of the real numbers and the multiplication of the real numbers.
Now, for some of the binary operations, we have what is called an identity element. This is a special element that leaves things alone under that particular operation. In addition, the identity is 0. Any number when added to 0 results in the same number. For the multiplication of numbers, the identity is 1. Any number multiplied by 1 results in the same number.
The matrix multiplication also contains an identity element. It is the matrix that leaves another matrix alone when it is multiplied by it. It acts just like the multiplication of the real numbers by 1.
2. How to find the determinant of a rectangular matrix such as a 2 x 3 matrix?
You cannot find the determinant of a rectangular matrix because determinant is only defined for the square matrices.
The determinant is said to represent the size of the n-dimensional hyperspace that is occupied by the n-dimensional rectangular parallelepiped which is having the column vectors of the square matrix in the form of its sides.
Thus, in the case of a wide rectangular matrix such as 2 x 3, it does not make any sense to define the determinant, since there are three 2-dimensional vectors that do not form a parallelogram when it is drawn from the origin in the cartesian coordinate system.
Also, in the case of the tall rectangular matrix such as 3 x 2 you have two 3-dimensional vectors that cannot form the sides of a parallelepiped.
